Finite Group Databasebeta

← #19758

Subgroups — #19758

Order 256.

368 conjugacy classes of subgroups.

The subgroup lattice has 325 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3334C211yesmaximal, characteristic, complemented
2#3294C211yesmaximal, characteristic, complemented
3#3283C211yesmaximal, characteristic, complemented
4#3272C211yesmaximal, characteristic, complemented
5#1722C211yesmaximal, characteristic, complemented
6#1689C211yesmaximal, characteristic, complemented
7#1681C211yesmaximal, characteristic, complemented
8#1673C211yesmaximal, characteristic, complemented
9#1665C211yesmaximal, characteristic, complemented
10#1647C211yesmaximal, characteristic, complemented
11#1647C211yesmaximal, characteristic, complemented
12#1639C211yesmaximal, characteristic, complemented
13#1625C211yesmaximal, characteristic, complemented
14#1581C211yesmaximal, characteristic, complemented
15#1580C211yesmaximal, characteristic, complemented
16C2×AGL(1,ℤ/8)—21no—
17C2×#135—21no—
18#547C2211yescharacteristic
19#547C2211yescharacteristic
20#547C2211yescharacteristic, complemented
21D42C2211yescharacteristic
22#500C2211yescharacteristic, complemented
23#500C2211yescharacteristic, complemented
24#494C2211yescharacteristic, complemented
25#492C2211yescharacteristic
26#486C2211yescharacteristic
27#486C2211yescharacteristic
28#480—21no—
29#480—21no—
30#480—21no—
31#474C2211yescharacteristic
32#474C2211yescharacteristic
33#474C2211yescharacteristic, complemented
34#468—21no—
35#467—21no—
36#466—21no—
37#466—21no—
38#461C2211yescharacteristic
39#461C2211yescharacteristic
40#461C2211yescharacteristic, complemented
41#459C2211yescharacteristic
42#459C2211yescharacteristic
43#452—21no—
44#452—21no—
45#450—21no—
46#449—21no—
47#447—21no—
48C4×D8C2211yescharacteristic
49C4×D8C2211yescharacteristic
50#436C2211yescharacteristic
51C8×D4C2211yescharacteristic
52C8×D4C2211yescharacteristic, complemented
53C8×D4C2211yescharacteristic, complemented
54C2×#107—21no—
55C2×#102—21no—
56#335C2211yescharacteristic, complemented
57#334C2211yescharacteristic, complemented
58#332C2211yescharacteristic, complemented
59#332C2211yescharacteristic, complemented
60#332C2211yescharacteristic, complemented
61#331C2211yescharacteristic
62D4⋊C8C2211yescharacteristic
63D4⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic, complemented
65D4⋊C8C2211yescharacteristic, complemented
66#322C2211yescharacteristic, complemented
67C2×#41—21no—
68C2×#41—21no—
69C2×#41—21no—
70C22×D4—21no—
71C22×D4—21no—
72AGL(1,ℤ/8)—44no—
73#135—44no—
74C2×Q16—21no—
75C2×Q16—21no—
76C2×SD16—21no—
77C2×SD16—21no—
78C2×SD16—21no—
79C2×SD16—21no—
80C2×SD16—21no—
81C2×D8—21no—
82C2×D8—21no—
83C2×D8D411yescharacteristic
84C2×D8D411yescharacteristic
85C2×M16—21no—
86C22×C8—21no—
87C22×C8—21no—
88C22×C8—21no—
89C4⋊Q8C2311yescharacteristic
90C4⋊Q8C2311yescharacteristic
91C4⋊Q8C2311yescharacteristic
92#127C2311yescharacteristic
93#123—41no—
94#123—41no—
95#123—41no—
96C22⋊Q8—21no—
97C22⋊Q8—21no—
98C22⋊Q8—21no—
99#121—21no—
100#121—21no—
101#121—21no—
102#121—21no—
103#121—21no—
104#120—41no—
105C4×D4C2311yescharacteristic
106C4×D4C2311yescharacteristic
107C4×D4C2311yescharacteristic
108C4×D4C2311yescharacteristic
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112#107—21no—
113#107—21no—
114#107—21no—
115#107—21no—
116#107—21no—
117#107—41no—
118#107D411yescharacteristic
119#107D411yescharacteristic
120C4⋊C8C2311yescharacteristic
121C4⋊C8C2311yescharacteristic
122C4⋊C8C2311yescharacteristic
123C4⋊C8C2311yescharacteristic
124#103—21no—
125#103—21no—
126#103—21no—
127#102—21no—
128#102—21no—
129#102—21no—
130#102—21no—
131#102—21no—
132#102—21no—
133#102—21no—
134#102—41no—
135#102D411yescharacteristic
136#102D411yescharacteristic
137#102D411yescharacteristic
138#102D411yescharacteristic
139C22⋊C8—21no—
140C22⋊C8—21no—
141C22⋊C8—21no—
142C22⋊C8—21no—
143C4×C8C2311yescharacteristic
144C4×C8C2311yescharacteristic
145C4×C8C2311yescharacteristic
146C24—41no—
147#41—42no—
148#41—42no—
149#41—42no—
150#41—81no—
151#41—81no—
152#41—81no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—41no—
160C2×D4—41no—
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—41no—
164C2×D4—41no—
165C2×D4—41no—
166C2×D4—81no—
167C2×D4—81no—
168C2×D4C2×D411yescharacteristic
169C2×D4C2×D411yescharacteristic
170C2×D4C2×D411yescharacteristic
171C2×D4C2×D411yescharacteristic
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179Q16—42no—
180Q16—81no—
181SD16—42no—
182SD16—42no—
183SD16—42no—
184SD16—42no—
185SD16—81no—
186D8—41no—
187D8—42no—
188D8—81no—
189D8D814yescomplemented
190M16—42no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—81no—
199C2×C8C2×D411yescharacteristic
200C2×C8C2×D411yescharacteristic
201C2×C8C2×D411yescharacteristic
202C2×C8C2×D411yescharacteristic
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4C2×D411yescharacteristic
213C4⋊C4C2×D411yescharacteristic
214C4⋊C4C2×D411yescharacteristic
215C4⋊C4C2×D411yescharacteristic
216C22⋊C4—21no—
217C22⋊C4—21no—
218C22⋊C4—21no—
219C22⋊C4—21no—
220C22⋊C4—41no—
221C22⋊C4—41no—
222C22⋊C4—41no—
223C22⋊C4—41no—
224C42C2411yescharacteristic
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—41no—
230C23—82no—
231C23—82no—
232C23—161no—
233Q8—41no—
234Q8—41no—
235Q8—41no—
236Q8—41no—
237Q8—41no—
238Q8—41no—
239D4—21no—
240D4—21no—
241D4—21no—
242D4—21no—
243D4—21no—
244D4—21no—
245D4—41no—
246D4—41no—
247D4—41no—
248D4—41no—
249D4—81no—
250D4—82no—
251D4—82no—
252D4—82no—
253D4—82no—
254D4—82no—
255D4C2×D812yes—
256D4C2×D812yes—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4C22×D411yescharacteristic
276C2×C4C22×D411yescharacteristic
277C2×C4H(5,𝔽2)11yescharacteristic
278C8—21no—
279C8—21no—
280C8—21no—
281C8—41no—
282C8—41no—
283C8—41no—
284C8—41no—
285C8—41no—
286C8C2×D812yes—
287C22—22no—
288C22—22no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—41no—
299C22—81no—
300C22—81no—
301C22—81no—
302C22—164no—
303C22D4211yescharacteristic
304C4—21no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—81no—
310C4—81no—
311C4—81no—
312C4C22×D811yescharacteristic
313C4#57511yescharacteristic
314C4#57611yescharacteristic
315C4#57711yescharacteristic
316C2—42no—
317C2—42no—
318C2—81no—
319C2—81no—
320C2—161no—
321C2#326711yescharacteristic
322C2D4×D811yescharacteristic
323C2#328911yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #3334C2semidirect
2. #3294C2semidirect
3. #3283C2semidirect
4. #3272C2semidirect
5. #1722C2semidirect
6. #1689C2semidirect
7. #1681C2semidirect
8. #1673C2semidirect
9. #1665C2semidirect
10. #1647C2semidirect
11. #1647C2semidirect
12. #1639C2semidirect
13. #1625C2semidirect
14. #1581C2semidirect
15. #1580C2semidirect
20. #547C22semidirect
22. #500C22semidirect
23. #500C22semidirect
24. #494C22semidirect
33. #474C22semidirect
40. #461C22semidirect
52. C8×D4C22semidirect
53. C8×D4C22semidirect
56. #335C22semidirect
57. #334C22semidirect
58. #332C22semidirect
59. #332C22semidirect
60. #332C22semidirect
64. D4⋊C8C22semidirect
65. D4⋊C8C22semidirect
66. #322C22semidirect
189. D8D8semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#33342
#32942
#32832
#32722
#17222
#16892
#16812
#16732
#16652
#16472
#16392
#16252
#15812
#15802